JP 2012-130108A discloses an example of a power supply device to be installed in a vehicle. The power supply device disclosed in Patent Document 1 includes a lead storage battery and a lithium storage battery, the power supply device being provided with an electrical supply line serving as a power path between the lead storage battery and the lithium storage battery. Also, two MOSFETs for switching conduction of the electrical supply line on and off are provided. This power supply device switches the MOSFETs on/off depending on the state of charge (SOC) of the lithium storage battery during, for example, non-regeneration (such as idle operation, acceleration traveling, or steady traveling) so as to perform control such that the SOC is within an optimum range.